A team of physicists and mathematicians has come up with a statistical technique that puts the fine details back into computer simulations of large-scale phenomena like air circulation in the atmosphere and currents in the ocean.

Computer models are generally good at capturing the big picture, but they are often forced to ignore things that happen at small scales. For example, models of a planet’s atmosphere capture the large-scale dynamics of jets and airflows, but they don’t include small-scale dynamics created by things like clouds and localized turbulence, despite the fact that those dynamics can often influence the larger scales.

“There are simply too many numbers for the computer to simulate it at a reasonable speed,” said Brad Marston, a Brown University physicist. “It might take years to simulate a day of the atmosphere, which wouldn’t be good.”

Simplify climate model simulations

The traditional approach to dealing with the problem is to simply lop the small scales off of the simulation. A few ad hoc ways of putting some of that information back in exist, but they tend not to be mathematically rigorous.

“These schemes have always suffered from the criticism that they lack predictive power,” Marston said. “You have to make a lot of decisions that you really shouldn’t have to make but you’re forced to make.”

In a paper published in the journal Physical Review Letters, Marston and his colleagues show a method of averaging out those small-scale dynamics in a way that is computationally tractable, which allows those dynamics to be simulated and their effects to be captured in a rigorous way.

“We’re retaining the degrees of freedom at the small scale, but treating them in a different way,” Marston said. “We don’t have to simulate all the little swirls, so to speak. We treat them by using their averages and the sizes of their fluctuations. It allows us to capture the contributions of these small-scale dynamics that would normally not be included.”

Climate model simulated air jets

In their paper, the researchers used the technique to model air jets forming on a round surface. They showed that the method produces results similar to brute-force numerical simulations of the same jets.

There have been prior attempts to treat small-scale disturbances statistically, Marston said, but those haven’t fared very well. Prior attempts have treated disturbances as being homogeneous and assumed they were not traveling in any one particular direction.

“But that almost never happens in nature,” Marston said. “Turbulence almost always has some directionality to it. That directionality is what makes these kinds of approximations work. It makes these approximations tenable.”

The researchers hope that the method might make for more accurate simulations of a wide variety of natural phenomena, from how the churning interiors of planets create magnetic fields to how air flows across the surfaces of cars or airplanes.

The method could be particularly useful in modeling Earth’s changing climate because the technique can more rigorously capture the influence of cloud formation.

“Cloud formation is seen as the largest source of uncertainty in climate models right now,” Marston said. “There are famous examples where different climate models that have different ways of dealing with the clouds give you qualitatively different results. In a warming world, one model might produce more clouds and another might produce fewer.”

By averaging those cloud dynamics and then simulating them in the models, it might be possible to reduce some of that uncertainty, Marston said.

The team has already started working to incorporate the method in climate simulations, as well as simulations of ocean currents and problems in astrophysics dealing with the behavior of plasmas.

“There are a whole bunch of problems out there where we feel this could be helpful,” Marston said.

Abstract

Quasilinear theory is often utilized to approximate the dynamics of fluids exhibiting significant interactions between mean flows and eddies. We present a generalization of quasilinear theory to include dynamic mode interactions on the large scales. This generalized quasilinear (GQL) approximation is achieved by separating the state variables into large and small zonal scales via a spectral filter rather than by a decomposition into a formal mean and fluctuations. Nonlinear interactions involving only small zonal scales are then removed. The approximation is conservative and allows for scattering of energy between small-scale modes via the large scale (through nonlocal spectral interactions). We evaluate GQL for the paradigmatic problems of the driving of large-scale jets on a spherical surface and on the beta plane and show that it is accurate even for a small number of large-scale modes. As GQL is formally linear in the small zonal scales, it allows for the closure of the system and can be utilized in direct statistical simulation schemes that have proved an attractive alternative to direct numerical simulation for many geophysical and astrophysical problems.